Suppression of Density Fluctuations in a Quantum Degenerate Fermi Gas

We study density profiles of an ideal Fermi gas and observe Pauli suppression of density fluctuations (atom shot noise) for cold clouds deep in the quantum degenerate regime. Strong suppression is observed for probe volumes containing more than 10 000 atoms. Measuring the level of suppression provides sensitive thermometry at low temperatures. After this method of sensitive noise measurements has been validated with an ideal Fermi gas, it can now be applied to characterize phase transitions in strongly correlated many-body systems.

Figure 1

Phase space diagram of ballistic expansion of a harmonically trapped Fermi gas. Ballistic expansion conserves phase space density and shears the initially occupied spherical area into an ellipse. In the center of the cloud, the local Fermi momentum and the sharpness of the Fermi distribution are scaled by the same factor, keeping the ratio of local temperature to Fermi energy constant. The same is true for all points in the expanded cloud relative to their corresponding unscaled in-trap points.

Figure 2

Comparison of density images to variance images. For Poissonian fluctuations, the two images at a given temperature should be identical. The variance images were obtained by determining the local density fluctuations from a set of 85 images taken under identical conditions. (a) Two dimensional image of the optical density of an ideal Fermi gas after 7 ms of ballistic expansion. The noise data were taken by limiting the field of view to the dashed region of interest, allowing for faster image acquisition. (b) For the heated sample, variance and density pictures are almost identical, implying only modest deviation from Poissonian statistics. (c) Fermi suppression of density fluctuations deep in the quantum degenerate regime manifests itself through the difference between density and variance picture. Especially in the center of the cloud, there is a large suppression of density fluctuations. The variance images were smoothed over $6\times 6$ bins. The width of images (b) and (c) is 2 mm.

Figure 3

Comparison of observed variances (black dots) with a theoretical model (black line) and the observed atom number (gray), at three different temperatures (a, b, and c), showing 50, 40, and 15% suppression. Noise thermometry is implemented by fitting the observed fluctuations, resulting in temperatures $T/T_F$ of $0.23\pm .01$, $0.33\pm .02$, and $0.60\pm .02$. This is in good agreement with temperatures $0.21\pm .01$, $0.31\pm .01$, and $0.6\pm .1$ obtained by fitting the shape of the expanded cloud [32]. The quoted uncertainties correspond to 1 standard deviation and are purely statistical.

Figure 4

Atom number variance vs average atom number. For each spatial position, the average atom number per bin and its variance were determined using 85 images. The filled and open circles in the figure are averages of different spatial bin positions with similar average atom number. For a hot cloud at $T/T_F=0.6$ (filled circles), the atom number variance is equal to the average atom number (dotted line, full Poissonian noise) in the spatial wings where the atom number is low. The deviation from the linear slope for a cold cloud at $T/T_F=0.21$ (open circles) is due to Pauli suppression of density fluctuations. There is also some suppression at the center of the hot cloud, where the atom number is high. The solid and dashed lines are quadratic fits for the hot and cold clouds to guide the eye.